I'm not sure about attaching GA to the page -- I will ping some LDN folks and see they can answer that better. I can help with your second question. Your assessment is correct -- Lithium's JS is stopping the event from bubbling up to your listener.
When you attach listeners with jQuery, you want to find the right balance between the element to attach and the selector to check. So here are two versions of your call, one at each extreme:
$(document).on("click", "#lia-body .lia-content .custom-search-wrapper .lia-autocomplete-container ul li a", ...);
This example attaches to the document, which is the top level. Almost all clicks anywhere on the document will bubble up to this level, and jQuery will then try to match the selector to see if it should handle the event. This is not ideal for performance, and it also means that any other listener can stop the event from bubbling up, so your listener may never run when you want it to. That's what's happening here -- two things you don't want.
$("#lia-body .lia-content .custom-search-wrapper .lia-autocomplete-container ul li a").on("click", null, ...);
This example does a query up front to find all the matching <a> elements on the page and then attaches a listener to each one individually. You get more listeners (more memory usage), and if any new <a>s enter the document after this, they won't have a listener on them, BUT there is a gain -- jQuery will only have to process click events on the <a>, not every click anywhere on the page. Finally, this makes your listener run earlier, which means it's less likely that some other event listener (e.g. Lithium's JS) will run first and get in the way.
Neither of these examples is probably ideal in most cases. The best balance usually would involve attaching to the most specific permanent element(s) on the page. So in this case, that might mean something like this:
$("#lia-body .lia-content .custom-search-wrapper .lia-autocomplete-container").on("click", "ul li a", ...);
If that still doesn't run, you might need to follow the second example and attach listeners directly on each <a> element, each time autocomplete results are generated (since they're new elements each time).
